Macroeconomic Dynamics Suvey Generalizations of Optimal Growth

1 Macroeconomic Dynamics Suvey Generalizations of optimal growth theory: stochastic models, mathematics, and meta-synthesis 1 2 Stephen Spear and Warren Young 1 Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213, [email protected] 2 Dept of Economics Bar Ilan University, 52900 Ramat Gan, Israel, [email protected] 2 Proposed Running Head: Stochastic Growth 3 Introduction In previous papers (Spear and Young 2014, 2015), we surveyed the origins, evolution and dissemination of optimal growth, two sector and turnpike models up to the early 1970s. Regarding subsequent developments in growth theory, a number of prominent observers, such as Fischer (1988), Stern (1991) and McCallum (1996) maintained that after significant progress in the 1950s and 1960s, economic growth theory "received relatively little attention for almost two decades" (Fischer, 1988, 329), and that "by the late 1960s early 1970s, research on the theory of growth more or less stopped" (Stern, 1991, 259). Stern went on to say "the latter half of the 1980s saw a rekindling of growth theory, particularly in the work of Romer… and Lucas" (1991, 259), that is to say, in the form of "endogenous growth" models. McCallum, for his part, wrote (1996, 41) "After a long period of quiescence, growth economics has in the last decade (1986-1995) become an extremely active area of research." Moreover, Brock and Mirman’s (1972) paper was the sole "extension" of Ramsey-Cass-Koopmans to a "stochastic environment" mentioned by McCallum (1996, 49). 4 This paper deals with the evolution of the "classical" growth research program of Ramsey-Cass-Koopmans vintage via its stochastic "variants" and "generalizations" (Samuelson 1976, note 1). Thus, here we trace the origins and impact of the stochastic generalization that brought about a paradigm shift in modern economics, and still generates significant research in the form of “quantitative macroeconomics”, that is to say, “real business cycle theory” (RBC henceforth), and its metamorphosis into the dynamic, stochastic general equilibrium (DSGE) approaches of both new Classical and new Keynesian vintage. The evolution of endogenous growth approaches and “New” and “Unified” growth models will be dealt with in a separate paper.1 Our focus, then, is on the origins and development of optimal stochastic growth models in continuous-time and discrete-time forms. The paper is divided into three sections. The first section deals with unpublished and published papers by Phelps (1960a, b; 1961; 1962a, b), and Mirrlees (1965a, b). Phelps' unpublished Cowles Foundation Papers on both continuous-time and discrete-time stochastic optimal growth (1960b, 1961) are also dealt with in this context—the former never published, the latter the basis for his 1962 Econometrica Paper. 1 It should be noted that the relatively new approach manifest in stochastic endogenous growth models will not be surveyed here, nor will the von Neumann-Gale model in its deterministic and stochastic versions; the evolution and development of these models will be dealt with elsewhere. 5 We then deal with Mirrlees' unpublished papers, dating from 1965, which had significant impact on subsequent work in the area of stochastic optimal growth, such as on the contributions of Merton, Mirman, and Brock and Mirman respectively. Mirrlees' use of the conceptual and mathematical tools provided by Wiener, Doob, and Ito is also dealt with, as they still influence the financial economics developed by Merton, based upon them. Merton's contributions (1969, 1975) to the continuous- time approach are also dealt with in this section. The second section deals with the application of the dynamic programming approach of Bellman and Blackwell over the period 1952- 1970, its application to economic planning and growth models, especially by Radner, over the period 1963-1974, and cross-fertilization between Radner , Brock and Mirman, and Radner's Ph.D student, Jeanjean. The third section tells the story of how Brock and Mirman developed their watershed approach over the period 1970-1973. It surveys the development of their 1972 JET (1972a) and 1973 IER papers from their origins in their early joint work and Mirman's thesis (1970), through conference presentation, and finally publication. This section also deals with the important, albeit little known third Brock-Mirman paper, that is, their 1971 conference paper published in the volume Techniques of Optimization (1972b). 6 1. Phelps, Mirrlees and Merton: unpublished and published papers, 1960-1975 Phelps: 1960-1962 In his June 1960 RAND paper "Optimal inventory policy for serviceable and reparable stocks", Phelps applied dynamic programming to the problem of determining "a unique stockage policy" regarding serviceable and reparable materials that would correspond to specific "decision regions". Phelps wrote that "the model developed to treat this sequential decision problem, in being one of the comparatively few two- dimensional dynamic programing models for which the structure of the optimal policy has been ascertained, may be of some methodological interest" (1960a, 4). He went on to apply Bellman's "principle of optimality" to "current" and "future decisions", such that "all future decisions must be optimal" so as to achieve "overall optimality" (1960a, 7). In dealing with "the infinite- stage program", he applied the "fundamental theorems of dynamic programming for decision processes" outlined by Bellman (1960a, 11). In December 1960, Cowles Foundation Discussion Paper [CFDP] 101 entitled "Capital risk and household consumption path: a sequential utility analysis" by Phelps appeared. The paper presented what Phelps called "a stochastic process of capital growth" (1960b, 1) in "a continuous 7 time formulation" (1961, 1). Phelps' CFDP 101 was never published. In our view, the paper is important in three respects. First, it was perhaps the earliest paper to apply an ostensibly continuous time approach to optimal stochastic growth, although Phelps' 1960 approach will be shown to be problematic, to say the least. Second, it was also one of the earliest papers to apply a dynamic programming approach and the Bellman (1957) "principle of optimality" to stochastic optimal growth (1960b, 9). Third, Phelps’ 1960 CFDP 101 provided the basis for the discrete-time extension of his approach in the form of his subsequent February 1961 CFDP 109, entitled "The accumulation of risky capital: a discrete-time sequential utility analysis". Now, while CFDP 101 (1960b) was cited by Phelps in his 1961 CFDP (1961, 1-3, 33), only CFDP 109 was mentioned by him in a note in his 1962 Econometrica paper, albeit with its title referred to incorrectly as being "identical" to the Econometrica paper (1962a, 733 note 6 ). This may explain why CDFP 101 has gone uncited and CFDP 109 sparsely-cited accordingly. But more is involved here than the fact that Phelps' CFDPs have gone virtually unnoticed until now. In correspondence regarding CFDP 101, Phelps wrote (23 September 2014): "I don't recall any reaction to CF 101 at all… I did not continue with the work that started in CF 101 8 because it was not clear to me that I had the analytical tools to push it any farther". With regard to the relationship between CFDP 101, CFDP 109 and his Econometrica 1962 paper, he wrote (23 September 2014): I did not really "switch" to the discrete-time framework. I had already done all or most of the work for it in my first post-doctoral job at the RAND Corporation 2. (The great Richard Bellman was there, as you very likely know, so I finally showed it to him. "That's trivial", he exclaimed. "The capital stock goes to infinity!" Of course the whole exercise was aimed at characterizing that path, solving for the consumption function, etc.) When to my surprise I ended right back at Yale in September 1960, I worked on the Golden Rule and what became CF 101. Then, frustrated by how hard CF 101 was, I prepared the discrete-time paper for what became the CF 109. 2 Phelps had utilized Bellman's dynamic programming approach in his RAND papers on "Optimal inventory policy for serviceable and replaceable stocks" (1960a), as indicated above, and in his paper "Optimal decision rules for procurement, repair or disposable spare parts" (1962b). 9 Turning first to Phelps' 1960 CDFP 101, a number of points stand out. Phelps described what he was analyzing as a "continuous time formulation" of a "stochastic process" (1960b, 8-10; 1961, 1). He then applied dynamic programming and the optimality principle (Bellman 1957) as the analytical basis for his approach (1960b, 9-17). As Phelps put it (1960b, 9, 16-17): "In what follows we take our inspiration from Chapter 9 of Bellman… our continuous-time process can be viewed as the limiting case of a discrete-time process in which the length of each period goes to zero while the number of discrete periods goes to infinity…" A close reading of Phelps' 1960 CFDP 101 reveals the difficulties Phelps faced in attempting to apply and develop his ostensibly "continuous" approach. First of all, he stated that in his model "capital gains and losses occur in unit amounts… fluctuations in the capital stock occur at random times… Thus capital grows according to a discontinuous Markov process" (1960b, 2) [our emphasis]. In other words this meant that the expected time path of wealth shocks was continuous, even though the underlying random shocks were discontinuous, that is to say, a "mixed model" rather than a pure continuous-time approach. 10 Second, the result Phelps outlined in his 1960 CFDP 101 was that a consumer could end up "ruined" (1960b, 3) due to "absorbing" states and "cyclic" states, although he noted that there could be both low and high capital persistence states that occur with positive probability, given the finite time horizon he assumed (1960b, 21-22). If, however, Phelps had pushed this through for an infinite horizon setting, he would have ended up with an ergodic distribution of the recurrent states, and the high or low capital "traps" (1960b, 21) would not occur, because there would be no absorbing states.

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